Fairly often, I'm asked to correct intonation on an acoustic guitar.
I believe this is rightly done at the bridge in the form of a compensated saddle,
where each stringís vibrating length can be set to achieve reasonable intonation.
There are much more sophisticated methods of achieving intonation correction; there
is more than one system of tempering the fretted scale. My interest is in simply
achieving the reasonable intonation that's found on a well-made conventional guitar,
without undue modification.

At this time, we generally accept modifications at the bridge because bridges are
relatively easy to replace. Some systems for correcting intonation require shortening
the fingerboard at the nut. This is not currently a generally accepted modification,
although it has merit from an engineering standpoint.

I check for intonation by the usual method of playing the string fretted at the 12th
fret and comparing to the note produced by playing the harmonic at that same position.
Most often, I'm reconfiguring a saddle or bridge to correct for a guitar that plays
sharp up the neck. Here's a formula I use to save the effort of trial-and-error.

Let's assume I'm working on a guitar that plays SHARP when fretted, and that all
other aspects of set up are satisfactory, e.g. string gauge and action. Iíll start
with the Low E, and repeat the procedure for all the strings

Compare intonation at 12th fret using an electronic tuner. Observe the NUMBER OF
CENTS sharp the fretted note is compared to the open string harmonic. IT PLAYS 8
CENTS SHARP (that's a lot, but I often see worse.)

ONE CENT IS ONE HUNDREDTH OF A SEMITONE. I think of one cent as ONE PERCENT. And,
I think of the number of cents error in intonation as the PERCENT ERROR. So my E
strings plays 8 PERCENT SHARP.

Therefore, If I know the LENGTH of a semitone, I can calculate the distance I must
move the pivot point of the string to correct for intonation.

My guitar has a scale length of 25-1/4Ē and I can look up the distance from the nut
to the center of the first fret on a fret scale chart, or I can simply measure it.
A SIMPLE MEASUREMENT IS ALL I NEED, because Iíll round off the decimal places, so
I measure 1.43 inches. (For my purposes, a measurement of 1-1/2 inches is probably
accurate enough to get reasonable results, but with my dial caliper I donít have
any trouble getting 2 decimal places.)

Here we go then: FIRST FRET DISTANCE times PERCENT ERROR
For my E string, itís 1.43Ē x 8% = 0.114Ē or a little more than 7/64Ē (a fair distance
when you think about it.)

I can now plot my ideal saddle positions for all the strings by starting with the
points where the strings cross the saddle. I can choose whether to compensate the
existing saddle by carving the top of it fore and aft, or by routing for a wider
saddle, or by inlaying the saddle slot and routing to relocate the saddle in the
bridge.

The advantage of this method is that it works easily for even the most bizarre instrument,
stringing, tuning, and setup combinations.

My biggest source of error in measuring is the intonation measurement with my electronic
tuner - you know how the meter wants to move around a bit. . .

Here's a simple explanation of the reasoning, submitted by Greg Neaga of Stuttgart,
Germany:

It is easy to think this through if you use a very extreme intonation flaw as
an example. Let's assume the pitch at the 12th fret is one complete semitone too
flat.

In this case, you would have to move the saddle towards the nut by distance equal
to the 1st fret distance.

Assuming the string tension stays the same, this would have the following effects:

1) The open string is raised by one semitone
2) The 12th fret harmonic is raised by one semitone
3) The fretted note at the 12th fret is raised by 2 semitones